Note $\triangle BGE\sim \triangle DGA$ and the ratio of their sides is $1:2$ (why?) . Use this to find $[ABE]$ in terms of $[BGE]$. Since $B$ is a midpoint, you can also find $[ABE]$ in terms of $[ABCD]$. Combining these two, you can find $[ABCD]$ in terms of $[BGE]$.

The correct answer is $12$.